The correct answer is "they."

This is a case of subject-pronoun disagreement. Although the subject in the first clause is "the budget," the subject becomes "the Board of Directors" in the second clause. Therefore, the pronoun must be replacing that subject, and the subject is singular ("the Board of Directors"), while "they" is plural, so it is a definite error. While there are several people, at least more than one, in a Board of Directors, the subject is not the directors, it is the entity itself— the Board. There is only one Board.

Though "was not decided" is in the passive voice, it is not grammatically wrong, just weaker writing. "By" is the correct usage, "and had"  is a verb-verb agreement, both are in past tense.

"Between Jenny and I" is incorrect. In this case, the sentence should say, "Lauren sulked all during school because she felt 'left out,' but the conversation was between Jenny and me, and I had every right to keep it that way." This sentence has a pronoun case error. Since Jenny and the speaker are objects (grammatically) within the sentence, the objective case pronoun ("me") is required, instead of the subjective case pronoun ("I"). In order to better understand these situations, when the phrase includes a noun and a pronoun simply remove the noun before the pronoun, for example, remove "Jenny and." Once you do that, you will much more easily identify the correct pronoun.

Another example: "Dad bought Roger and I a bicycle." As opposed to: "Dad bought (Roger and) I a bicycle." One would not say: "Dad bought I a bicycle."

In this sentence, the pronoun “it” does not match in number with its antecedent “Members.” The corrected sentence reads: Members of the blue team travel around the city collecting buttons, and they will not be able to communicate with the red team at any time.

The error in this sentence arises where we are told that Bianca and her sister disagree over their parents' divorce, but her sister is presented abstractly; the correct usage of "her sister" allows the individual mentioned to be presented in relation to Bianca. Without the "her," we cannot be certain what the relation is, although it is clearly indicated elsewhere in the sentence.

This sentence has an ambiguous pronoun. To whom does "she" refer? You could fix this either by referring to both women with "they," if it is indeed the case that both were honored at the awards ceremony after dinner, or by replacing "she" with the name of the woman being specified.

This sentence contains an ambiguous pronoun. We don't know whether "him" refers to Mr. Harris or Mr. Jarvins, so "him" is the error in the sentence. If we wanted to fix the error, we would have to replace "him" with the unique name of the person—Mr. Harris or Mr. Jarvins—being considered for the promotion.

This is an ambiguous pronoun. We are not sure to whom "her" refers. The sentence should use her name.

"They" is an ambiguous pronoun reference—it could refer to either "experts" or "children."

"She" contains this sentence's error. As Susanne and Patricia are both girls, the pronoun "she" is ambiguous. Also, the "she" could refer to another person—perhaps someone observing the competition in the audience. We don't know if the she refers to Susanne, Patricia, or someone else, and this makes the sentence's meaning unclear.

While the subjects of the sentence are "Jane and Annabel," the sentence later uses the pronoun "she," which could refer to either of the women. We can correct this ambiguity by replacing "she" with "they."